Abstract: The integers that are a sum of two squares are characterized by the classical theorem of Fermat as those with a prime factorization in which every prime congruent 3 mod 4 appears with even multiplicity. A theorem of Landau determines the asymptotic behavior of the number of such integers up to x, but their behavior in short intervals is not fully understood. I will report on joint work with Efrat Bank and Lior Bary-Soroker in which we address a function field analogue of the latter problem in the large finite field limit.
Time 14:30-15:30,
Location: Mathematics Building, Lecture Hall 2, Givat Ram (Jerusalem)
Speaker: Terence Tao (UCLA)
Abstract: I will discuss the mean square of sums of the generalised divisor function over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q tends to infinity we establish a relationship with a matrix integral over the unitary group, and analyse the integral. This is a joint work with Jon Keating, Brad Rodgers and Zeev Rudnick. I will also discuss the auto-correlations of arithmetic functions (in particular the generalised divisor function and the von Mangoldt function). Function field analogues of these problems have recently been resolved in the limit of large finite field size q. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. We compute averages of terms of lower order in q which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when q tends to infinity. This is a joint work with Jon Keating.
Abstract: The talk will be an introduction on Farey sequences and their basic properties. Time permitting, I will prove some other results, such as uniform distribution and their limiting gap distribution.
Abstract: A basic concern in number theory is additive number theory, aiming to find integer solutions to some Diophantine equations. In this direction, seeking prime solutions to the Diophantine equation form an interesting research field. The Waring-Goldbach problem seeks to represent integers by powers of primes. The twin prime conjecture and the Goldbach problem are just linear examples of the Waring-Goldbach problem. The circle method of Hardy and Littlewood in combination with the estimates of Vinogradov for exponential sums over primes gives an affirmative answer to the general Waring-Goldbach problem. In this talk, I will give an overview of the Waring-Goldbach problem and introduce some recent progresses concerning the circle method.
Abstract: Given a polynomial map f of the plane into itself and a "starting point" P, its forward orbit under f is the set { P, f(P), f(f(P)),...}. If the orbit intersects a curve C infinitely many times, what can be said about the curve C? The dynamical Mordell-Lang conjecture asserts that this can only happen for "the obvious reason", namely that C is f-periodic ("land once, and you land infinitely often") if the curve is algebraic and irreducible. We will give some background on the conjecture, and using a p-adic analytic approach, prove it in certain cases. If the modulo p reduction of f is "sufficiently random" the approach should work for quite general maps. However, in sufficiently high dimensions the approach breaks down due to the modulo p periods being too long (it is then impossible to avoid a sparse set of "bad points".) We will discuss this in more detail, and present numerical evidence for the validity of the random map assumption in various dimensions.
Abstract: Given the intersection of n hypersurfaces of fixed degrees d_{1},...,d_{n} in the projective n-space defined by random equations over a large finite field one can consider the Frobenius action on their intersection. We show that the statistics of the cycle structure of this action agrees with the statistics of a random permutation on d_{1},...,d_{n} elements.
Abstract: We study some problems in the representation of integers as sums of three squares and their applications to the theory of random waves on the three-dimensional torus.
Abstract: In the last decade Apollonian circle packings have aroused interest from both number theorist and homogeneous dynamicists. In this talk I will discuss the arithmetic part of the Apollonian circle packings, and its generalization to other types of circle packings.
Homework assignment, due November 6, 2014
Abstract: There are few known transcendence results for polylogarithmic values except for Zeta(2k). The talk will be about construction of rational approximations of polylogarithm functions and also on measures of irrationality of these numbers. An important part of this talk is related to Fuchsian differential equations and special functions.
Homework assignment, due November 20, 2014
Abstract: A classical conjecture of Goldston and Montgomery estimates the variance in counts of primes in short intervals. This quantity, they showed, is related to the local distribution of zeros of the Riemann zeta function. We put forward a generalization of this conjecture for almost-primes (integers comprised of a fixed number of prime factors). The generalization is interesting in part because of its surprisingly simple algebraic character. We will also discuss analogous results in a function field setting, following work of Katz and Keating and Rudnick, who proved a function field analogue of the conjecture of Goldston and Montgomery. A key role is played by the analysis of certain random matrix integrals.
Abstract: Modular forms appear prominently in number theory and also come into play in other areas of mathematics as well as physics. In this talk I will mention some basic properties of modular forms and describe the distribution of their zeros. In particular, I will discuss how Quantum Unique Ergodicity for Hecke cusp forms is related to the equidistribution of their zeros.
Abstract: The prime number theorem for arithmetic progressions uses properties of Dirichlet characters, and their respective L-functions, to prove the equidistribution of primes across arithmetic progressions. In this talk I will explain how this same approach can be modified to prove the equidistribution of Gaussian primes across sectors of the complex plane, by making use of the properties of (infinite) Hecke characters and their associated L-functions.
Abstract: In this talk I will discuss Landau's theorem about the number of sums of two squares and some of the problems arising from it.
Homework assignment 3, due December 11, 2014
Abstract: In the talk, I plan to discuss and sketch the proofs of several classical results that concern
(a) Taylor series with integer coefficients;
(b) Entire functions, which attain integer values at non-negative integers;
(c) Algebraic numbers lying together with all conjugates in a given compact set of "a relatively small size".
Abstract: Linnik proved in the late 1950's the equidistribution of integer points on large spheres of the 3-dimensional space, under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec). We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition. Using unipotent dynamics we obtain stronger equidistribution results of the higher dimensional analogs. I will present these arithmetic problems in detail, their translation into dynamics on homogeneous spaces and then discuss the tools which enable to prove the corresponding dynamical result. Prerequisites will be kept to a minimum; the dynamical result is on S-arithmetic homogeneous spaces so in the second part of the talk p-adic numbers will be assumed.
Abstract: A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice Z^{2}, gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be "attainable" from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a ``fractal'' structure. This complicated structure in some sense arises from prime powers - singularities do not occur for circles of radius n^{1/2} if n is square free. This work is joint with Par Kurlberg.
Homework assignment 4, due January 15, 2015
Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 640-7806